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A323223 a(n) = [x^n] x/((1 - x)*(1 - 4*x)^(5/2)).

%I A323223 #17 Jan 29 2019 09:10:42
%S A323223 0,1,11,81,501,2811,14823,74883,366603,1752273,8218733,37964449,
%T A323223 173172249,781607349,3496163949,15517771749,68412846069,299828796219,
%U A323223 1307168814519,5672308893819,24511334499219,105519144602439,452695473616239,1936085243038839,8256615564926439
%N A323223 a(n) = [x^n] x/((1 - x)*(1 - 4*x)^(5/2)).
%H A323223 Seiichi Manyama, <a href="/A323223/b323223.txt">Table of n, a(n) for n = 0..1653</a>
%F A323223 a(n) = ((5*n + 1)*a(n-1) - (4*n + 2)*a(n-2))/(n - 1) for n >= 2.
%F A323223 a(n) = -(-4)^n*binomial(-5/2, n)*hypergeom([1, n+5/2], [n+1], 4) - i*sqrt(3)/27.
%F A323223 a(n) ~ 2^(2*n+2) * n^(3/2) / (9*sqrt(Pi)). - _Vaclav Kotesovec_, Jan 29 2019
%F A323223 a(n+1) - a(n) = A002802(n). - _Seiichi Manyama_, Jan 29 2019
%p A323223 A323223List := proc(len) local ogf, ser; ogf := (1 - 4*x)^(-5/2)*x/(1 - x);
%p A323223 ser := series(ogf, x, (n+1)*len+1); seq(coeff(ser, x, j), j=0..len) end:
%p A323223 A323223List(24);
%p A323223 # Alternative:
%p A323223 a := proc(n) option remember; `if`(n<2,n,((5*n+1)*a(n-1)-(4*n+2)*a(n-2))/(n-1)) end: seq(a(n), n=0..24);
%Y A323223 Row 5 of A323222.
%Y A323223 Cf. A002802.
%K A323223 nonn
%O A323223 0,3
%A A323223 _Peter Luschny_, Jan 26 2019